Tauberian constants associated to centered translation invariant density bases

This paper provides a necessary and sufficient condition on Tauberian constants associated to a centered translation invariant differentiation basis so that the basis is a density basis. More precisely, given $x in mathbb{R}^n$, let $mathcal{B} = cup_{x in mathbb{R}^n} mathcal{B}(x)$ be a collection of bounded open sets in $mathbb{R}^n$ containing $x$. Suppose moreover that these collections are translation invariant in the sense that, for any two points $x$ and $y$ in $mathbb{R}^n$ we have that $mathcal{B}(x + y) = {R + y : R in mathcal{B}(x)}.$ Associated to these collections is a maximal operator $M_{mathcal{B}}$ given by $$M_{mathcal{B}}f(x) :=sup_{R in mathcal{B}(x)} frac{1}{|R|} int_R |f|.$$ The Tauberian constants $C_{mathcal{B}}(alpha)$ associated to $M_{mathcal{B}}$ are given by $$C_{mathcal{B}}(alpha) :=sup_{E subset mathbb{R}^n atop 0 < |E| < infty} frac{1}{|E|}|{x in mathbb{R}^n :, M_{mathcal{B}}chi_E(x) > alpha}|.$$ Given $0 < r < infty$, we set $mathcal{B}_r(x) :={R in mathcal{B}(x) : mathrm{diam } R < r}$, and let $mathcal{B}_r :=cup_{x in mathbb{R}^n} mathcal{B}_r (x).$ We prove that $mathcal{B}$ is a density basis if and only if, given $0 < alpha < infty$, there exists $ r = r(alpha) >0$ such that $C_{mathcal{B}_r}(alpha) < infty$. Subsequently, we construct a centered translation invariant density basis $mathcal{B} = cup_{x in mathbb{R}^n} mathcal{B}(x)$ such that there does not exist any $0 < r$ satisfying $C_{mathcal{B}_{r}}(alpha) < infty$ for all $0 < alpha < 1$.